Calculate the Product: 5^5 × 6^5 Power Multiplication

Q: Why can I write the same expression in two different ways?

Because multiplication is commutative! This means $ 5 \times 6 = 6 \times 5 $, so $ (5 \times 6)^5 $ and $ (6 \times 5)^5 $ are mathematically identical.

Q: How do I know when to use the power of a product rule?

Use it when you see same exponents on different bases being multiplied, like $ 5^5 \times 6^5 $. The rule lets you combine: $ a^n \times b^n = (a \times b)^n $.

Q: What if the exponents were different, like 5³ × 6⁵?

Then you cannot use the power of a product rule! The exponents must be the same. You'd have to calculate each power separately and then multiply.

Q: Is (5×6)⁵ easier to calculate than 5⁵ × 6⁵?

Yes! $ (5 \times 6)^5 = 30^5 $ means you only calculate one power instead of two separate powers and then multiplying them together.

Q: Can I use this rule backwards too?

Absolutely! You can go from $ (a \times b)^n $ to $ a^n \times b^n $. This is called expanding the power, and it's very useful for simplifying expressions.

Power Rule Application with Product Equivalence

Choose the expression that corresponds to the following:

$5^5\times6^5=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 When we are presented with a multiplication where each factor has the same exponent (N)

00:07 The entire multiplication can be written with the exponent (N)

00:11 We will apply this formula to our exercise

00:22 This is one potential solution, now let's review another possible solution

00:25 In multiplication, the order of factors doesn't matter, therefore the expressions are equal

00:28 We will apply this formula to our exercise

00:36 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the expression that corresponds to the following:

$5^5\times6^5=$

Step-by-step solution

We are given the expression $5^5 \times 6^5$ .
According to the power of a product rule, $(a \times b)^n = a^n \times b^n$ . Therefore, we can rewrite the expression as $(5 \times 6)^5$ or $(6 \times 5)^5$ .

Both expressions $(5 \times 6)^5$ and $(6 \times 5)^5$ are mathematically equivalent. Therefore, the solution to the problem is that answers (a) and (b) are correct.

Final Answer

Answers (a) and (b) are correct.

Key Points to Remember

Essential concepts to master this topic

Power Rule: $a^n \times b^n = (a \times b)^n$ allows factoring
Technique: Convert $5^5 \times 6^5$ to $(5 \times 6)^5$ or $(6 \times 5)^5$
Check: Both $(5 \times 6)^5$ and $(6 \times 5)^5$ equal $30^5$ ✓

Common Mistakes

Avoid these frequent errors

Thinking only one form is correct
Don't choose between $(5 \times 6)^5$ and $(6 \times 5)^5$ = missing that both are equal! Multiplication is commutative, so the order doesn't matter. Always recognize that $5 \times 6 = 6 \times 5$ makes both expressions identical.

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can I write the same expression in two different ways?

Because multiplication is commutative! This means $5 \times 6 = 6 \times 5$ , so $(5 \times 6)^5$ and $(6 \times 5)^5$ are mathematically identical.

How do I know when to use the power of a product rule?

Use it when you see same exponents on different bases being multiplied, like $5^5 \times 6^5$ . The rule lets you combine: $a^n \times b^n = (a \times b)^n$ .

What if the exponents were different, like 5³ × 6⁵?

Then you cannot use the power of a product rule! The exponents must be the same. You'd have to calculate each power separately and then multiply.

Is (5×6)⁵ easier to calculate than 5⁵ × 6⁵?

Yes! $(5 \times 6)^5 = 30^5$ means you only calculate one power instead of two separate powers and then multiplying them together.

Can I use this rule backwards too?

Absolutely! You can go from $(a \times b)^n$ to $a^n \times b^n$ . This is called expanding the power, and it's very useful for simplifying expressions.

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